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A generalized definition of the fractional derivative with applications

A generalized fractional derivative (GFD) definition is proposed in this work Fora differentiable function expanded by a Taylor series, we show that (DD beta)-D-alpha f (t) = D alpha+beta f (t); 0 < alpha <= 1; 0 < beta <= 1. GFD is applied for some functions to investigate that the GFD...

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Main Authors: Abu-Shady, M., Kaabar, Mohammed K. A.
Format: Article
Published: Mathematical Problems in Engineering 2021
Subjects:
QA Mathematics
Online Access:http://eprints.um.edu.my/35329/
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spelling my.um.eprints.353292022-10-27T06:14:01Z http://eprints.um.edu.my/35329/ A generalized definition of the fractional derivative with applications Abu-Shady, M. Kaabar, Mohammed K. A. QA Mathematics A generalized fractional derivative (GFD) definition is proposed in this work Fora differentiable function expanded by a Taylor series, we show that (DD beta)-D-alpha f (t) = D alpha+beta f (t); 0 < alpha <= 1; 0 < beta <= 1. GFD is applied for some functions to investigate that the GFD coincides with the results from Caputo and Riemann-Liouville fractional derivatives. The solutions of the Riccati fractional differential equation are obtained via the GFD. A comparison with the Bernstein polynomial method (BPM), enhanced homotopy perturbation method (EHPM), and conformable derivative (Cl)) is also discus sal. Our results show that the proposed definition gives a much better accuracy than the wellknown definition of the conformable derivative. Therefore, GFD has advantages in comparison with other related definitions. This work provides a new path for a simple tool for obtaining analytical solutions of many problems in the context of fractional calculus. Mathematical Problems in Engineering 2021-10-23 Article PeerReviewed Abu-Shady, M. and Kaabar, Mohammed K. A. (2021) A generalized definition of the fractional derivative with applications. Mathematical Problems in Engineering, 2021. ISSN 1024-123X, DOI https://doi.org/10.1155/2021/9444803 <https://doi.org/10.1155/2021/9444803>. 10.1155/2021/9444803
institution Universiti Malaya
building UM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Malaya
content_source UM Research Repository
url_provider http://eprints.um.edu.my/
topic QA Mathematics
spellingShingle QA Mathematics
Abu-Shady, M.
Kaabar, Mohammed K. A.
A generalized definition of the fractional derivative with applications
description A generalized fractional derivative (GFD) definition is proposed in this work Fora differentiable function expanded by a Taylor series, we show that (DD beta)-D-alpha f (t) = D alpha+beta f (t); 0 < alpha <= 1; 0 < beta <= 1. GFD is applied for some functions to investigate that the GFD coincides with the results from Caputo and Riemann-Liouville fractional derivatives. The solutions of the Riccati fractional differential equation are obtained via the GFD. A comparison with the Bernstein polynomial method (BPM), enhanced homotopy perturbation method (EHPM), and conformable derivative (Cl)) is also discus sal. Our results show that the proposed definition gives a much better accuracy than the wellknown definition of the conformable derivative. Therefore, GFD has advantages in comparison with other related definitions. This work provides a new path for a simple tool for obtaining analytical solutions of many problems in the context of fractional calculus.
format Article
author Abu-Shady, M.
Kaabar, Mohammed K. A.
author_facet Abu-Shady, M.
Kaabar, Mohammed K. A.
author_sort Abu-Shady, M.
title A generalized definition of the fractional derivative with applications
title_short A generalized definition of the fractional derivative with applications
title_full A generalized definition of the fractional derivative with applications
title_fullStr A generalized definition of the fractional derivative with applications
title_full_unstemmed A generalized definition of the fractional derivative with applications
title_sort generalized definition of the fractional derivative with applications
publisher Mathematical Problems in Engineering
publishDate 2021
url http://eprints.um.edu.my/35329/
_version_ 1748181076950384640
score 13.244367

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